ISOLATED BOUNDARY SINGULARITIES OF SEMILINEAR ELLIPTIC EQUATIONS

Marie-Françoise Bidaut-Veron (1) , Augusto C. Ponce (2) , Laurent Veron (1)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique
2 Département de mathématique [Louvain]

Given a smooth domain $\Omega\subset\RR^N$ such that $0 \in \partial\Omega$ and given a nonnegative smooth function $\zeta$ on $\partial\Omega$, we study the behavior near $0$ of positive solutions of $-\Delta u=u^q$ in $\Omega$ such that $u = \zeta$ on $\partial\Omega\setminus\{0\}$. We prove that if $\frac{N+1}{N-1} < q < \frac{N+2}{N-2}$, then $u(x)\leq C \abs{x}^{-\frac{2}{q-1}}$ and we compute the limit of $\abs{x}^{\frac{2}{q-1}} u(x)$ as $x \to 0$. We also investigate the case $q= \frac{N+1}{N-1}$. The proofs rely on the existence and uniqueness of solutions of related equations on spherical domains.

Mots clés: Elliptic operator. Liouville theorem. A priori estimates. Asymptotic behaviour

Domaines: Equations aux dérivées partielles [math.AP]

Marie-Françoise Bidaut-Veron, Augusto C. Ponce, Laurent Veron. ISOLATED BOUNDARY SINGULARITIES OF SEMILINEAR ELLIPTIC EQUATIONS. 2009. ⟨hal-00358178v3⟩